What are the domain and range of the function?
f(x)=35×5
The Correct Answer and Explanation is:
To determine the domain and range of the function ( f(x) = 35x^5 ), we must understand the structure of the function and its behavior.
Domain:
The domain of a function consists of all the possible input values (x-values) that will result in a valid output for the function. The function ( f(x) = 35x^5 ) is a polynomial, where ( 35x^5 ) is a product of a constant (35) and a power of ( x ) (specifically, ( x^5 )).
Polynomials like ( 35x^5 ) are defined for all real numbers, meaning there are no restrictions or undefined values (like division by zero or square roots of negative numbers). Therefore, the domain of ( f(x) = 35x^5 ) is all real numbers.
Thus, the domain is:
[
\text{Domain} = (-\infty, \infty)
]
Range:
The range of a function consists of all the possible output values (f(x)-values) that the function can produce.
For ( f(x) = 35x^5 ), the function is a polynomial function of odd degree (degree 5). Let’s analyze how the output behaves as ( x ) changes:
- When ( x ) is positive, ( x^5 ) is also positive, so ( f(x) ) will be positive.
- When ( x ) is negative, ( x^5 ) is negative (since raising a negative number to an odd power results in a negative number), so ( f(x) ) will be negative.
- As ( x ) approaches infinity (( x \to \infty )), ( f(x) ) also goes to infinity (( f(x) \to \infty )).
- As ( x ) approaches negative infinity (( x \to -\infty )), ( f(x) ) goes to negative infinity (( f(x) \to -\infty )).
Therefore, the function ( f(x) = 35x^5 ) can take any real value, meaning its range is also all real numbers.
Thus, the range is:
[
\text{Range} = (-\infty, \infty)
]
Conclusion:
- The domain of ( f(x) = 35x^5 ) is ( (-\infty, \infty) ).
- The range of ( f(x) = 35x^5 ) is ( (-\infty, \infty) ).
This is because polynomial functions, particularly those with odd degrees, have domains and ranges that cover all real numbers.